274 XIV INTEGRATION IN LOCALLY COMPACT GROUPS For example, let us verify the first of the formulas (14.6.1.1). We have to show that, for every function fe ^C(G)? tne continuous function (x,y) ^f(xy) is (gs®ju)-integrable. Now the function x\-+f(xy) is es-integrable, and [f(xy)des(x) =f(sy). Since the function y*-*f(sy) is continuous and has compact support, it is ju-integrable. Our assertion therefore follows from "(13.21.10), and then the theorem of Lebesgue-Fubini (13.21.7) gives \[f(xy) dss(x) dn(y) = \f(sy) dp(y). The formula to be proved follows now from (14.1.1) and (14.1.2). (14.6.2) Every finite family (//1?..., jj,n) of bounded measures on G is con- volvable. The measure /^ * n2 * " " * f*n *s bounded, and (14.6.2.1) llA«i*A«2*'"*A.II ^ llA*ill ' llMill •"llA.II- To prove the first assertion, observe that the measure /^ ® • • • ® /zrt on G" is bounded (13.21.18). For every function /e Jfc(G), the function (xl9..., xn) \-^f(xix2 "• xn) is continuous and bounded on Gn, and therefore (/*! ® • • • ® /ij-integrable (13.20.4). To prove (14.6.2.1), it is enough to remark that if/e ^TC(G) and ||/|| ^ 1, then ! ll/*iII ' 1102II •" II0J by virtue of (13.21.18). (14.6.3) ^4 /e// /faar measure A on G is conceivable on the right with any bounded measure n on G, and \JL * A = We may restrict ourselves to the case where /f g; 0 (cf. (14.7.1 .2)). For each function /€ «^f+(G) we have J d/i(*) J and therefore, by virtue of (13.21.9), the function (x,y)\-*f(xy) is integrable and its integral is equal to A(/) ||/i||. The same calculation shows that X is not convotvable with itself if Q is not compact, for then the function 1 is not A-integrable (14.2.3). (14.6.4) Let (fa, . . . , /*„) be a finite sequence of measures on G, all of which except possibly for one have compact support. Then the sequence (jux, . . . , JUM) is convolvable. (Problem 6(c)). A subset A of Nc is relatively compact in Nc if and only if it satisfies